New classes of compact-type spaces

Abstract

Being motivated by the notions of $\hat{I}º$-Fréchet--Urysohn spaces and $k^{\prime }$-spaces introduced by Arhangel'skii, the notion of sequential spaces and the study of Ascoli spaces, we introduce three new classes of compact-type spaces. They are defined by the possibility to attain each or some of boundary points $x$ of an open set $U$ by a sequence in $U$ converging to $x$ or by a relatively compact subset $A\subseteq U$ such that $x\in \overline{A}$. Relationships of the introduced classes with the classical classes (as, for example, the classes of $\hat{I}º$-Fréchet--Urysohn spaces, (sequentially) Ascoli spaces, $k_{\mathbb{R}}$-spaces, $s_{\mathbb{R}}$-spaces etc.) are given. We characterize these new classes of spaces and study them with respect to taking products, subspaces and quotients. In particular, we give new characterizations of $\hat{I}º$-Fréchet--Urysohn spaces and show that each feathered topological group is $\hat{I}º$-Fréchet--Urysohn. We describe locally compact abelian groups which endowed with the Bohr topology belong to one of the aforementioned classes. Numerous examples are given.